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Bethe lattice

The Bethe lattice is an infinite, connected, acyclic graph in which every vertex has the same finite degree z \geq 2, known as the coordination number.^[1] Introduced by physicist Hans Bethe in 1935, it emerged from his analysis of atomic ordering in binary alloys, where the structure facilitates a mean-field approximation that accounts for short-range correlations without loops.^[2] Distinct from the finite Cayley tree, which features a root vertex with degree z and surface boundaries that introduce inhomogeneities, the Bethe lattice is homogeneous, extending infinitely without boundaries or cycles, ensuring a unique path between any two vertices.^[3] This tree-like topology, with exponentially growing shells of sites (N_s = z(z-1)^{s-1} for shell s), yields an infinite dimensionality and finite correlation length, decaying as \xi = -1 / \ln((z-1)/z).^[3] In statistical mechanics, the Bethe lattice enables exact solutions for models like the Ising ferromagnet, where phase transitions occur at critical temperatures kT_c / J = \frac{2}{ \ln \frac{z}{z-2} } for z > 2, and percolation, with threshold p_c = 1/(z-1).^[4]^[1] Its loop-free nature approximates higher-dimensional lattices by neglecting long-range correlations, providing benchmarks for phenomena such as magnetism, localization in disordered systems, and belief propagation algorithms in computer science.^[3]

History and definition

Historical development

The Bethe lattice concept originated in the work of Hans Bethe, who introduced it in 1935 as part of an approximation method for analyzing the statistical mechanics of binary alloys. In his paper on the statistical theory of superlattices, Bethe addressed the arrangement of atoms in alloys, where atoms of two types occupy lattice sites and form ordered structures at low temperatures. To account for local configuration fluctuations neglected in prior mean-field approaches like those of Bragg and Williams, Bethe employed a cluster expansion that effectively modeled the lattice as a loop-free structure, allowing recursive calculations of partition functions and order parameters. This tree-like approximation provided a more accurate treatment of short-range correlations while simplifying the problem to an infinite regular tree with a fixed coordination number.^[2] Early applications of the Bethe lattice focused on its utility as an infinite regular tree to model physical systems without cycles, enabling exact solutions for certain lattice models in statistical mechanics. By eliminating loops, the structure facilitated analytical progress in problems involving nearest-neighbor interactions, such as alloy ordering and ferromagnetism, where boundary effects could be ignored in the thermodynamic limit. This loop-free nature made it particularly suitable for approximate solutions that bridged mean-field theory and more exact methods, highlighting phase transitions through recursive relations for site occupancies or spins.^[5] During the 1950s and 1970s, the Bethe lattice gained prominence in studies of phase transitions, serving as a testbed for mean-field-like approximations on tree graphs that improved upon classical theories by incorporating partial correlation effects. Researchers applied it to models like the Ising ferromagnet, deriving critical temperatures and exponents via iterative methods that captured the onset of long-range order without the complexities of cyclic lattices. These developments underscored the lattice's role in understanding critical phenomena, with exact solvability on trees providing insights into universality classes and approximation hierarchies in statistical mechanics.^[5]

Formal definition

The Bethe lattice is formally defined as an infinite, connected, and acyclic graph in which every vertex has the same fixed degree q+1, where q \geq 1 is an integer parameter referred to as the order or branching factor of the lattice.^[6] The coordination number z is accordingly z = q + 1, denoting the uniform number of edges incident to each vertex.^[7] This structure guarantees the absence of cycles, rendering the Bethe lattice a tree in the graph-theoretic sense, with paths between any two vertices being unique.^[8] In its rooted formulation, the Bethe lattice originates from a central vertex, termed the root, which possesses degree q. From this root, q edges extend to the first generation of vertices, each of which has degree q+1: one edge connects to the parent (toward the root), while the remaining q edges branch to child vertices in the subsequent generation.^[2] This process continues indefinitely across successive generations, yielding a radially symmetric, loop-free hierarchy that expands outward without bound.^[9] The unrooted Bethe lattice, by contrast, is a homogeneous infinite tree in which all vertices are equivalent, each having precisely degree q+1, with no designated root or peripheral boundary.^[10] This version exhibits full vertex-transitivity, ensuring that the local neighborhood around any vertex mirrors that of any other, free from edge or asymmetry effects.^[11] Distinct from the finite Cayley tree, which serves as a bounded approximation with a root of degree q and terminal leaves introducing surface influences, the Bethe lattice emerges as the infinite-size limit of such trees, eliminating all boundary artifacts and enabling precise analytical treatments.^[9] It is frequently denoted in the literature as T_{q+1}, signifying the infinite (q+1)-regular tree.

Basic properties

Tree structure and coordination number

The Bethe lattice is an infinite connected acyclic graph, forming a tree structure with no cycles or closed loops.^[12] This absence of cycles ensures that the graph is loopless, facilitating recursive computations in models like the Ising model by eliminating cyclic dependencies. The lattice exhibits perfect regularity, where every vertex has the same fixed degree, denoted as the coordination number z \geq 2, an integer representing the number of neighboring vertices per site.^[1] In the standard unrooted formulation, the branching factor is z-1 \geq 1; for z-1 = 1 (z = 2), the structure is an infinite path, while z-1 = 2 (z = 3) yields a binary tree. In rooted constructions of the infinite Bethe lattice, every vertex has degree z: the root has z children, and each non-root vertex has one parent and z-1 children. As a tree, the Bethe lattice guarantees a unique path between any pair of distinct vertices, with the geodesic distance measured by the edge count along this path, often aligned with generational levels in rooted views.^[1] Unlike finite trees, the infinite Bethe lattice possesses no finite boundary, extending perpetually with its "boundary" conceptualized at infinity, which distinguishes it from bounded approximations like the Cayley tree.^[13]

Sizes of generations

In a rooted Bethe lattice with coordination number z \geq 2, the vertices are partitioned into generations based on their graph distance from the root vertex. The root forms generation 0 and contains N_0 = 1 vertex.^[14] Generation 1 consists of the z direct neighbors of the root, so N_1 = z. For n \geq 2, each vertex in generation n-1 connects to z-1 new vertices in generation n (accounting for the connection to its parent), yielding the recurrence N_n = (z-1) N_{n-1} and the closed form N_n = z (z-1)^{n-1}.^[14]^[6] The total number of vertices up to and including generation n is given by

S_n = 1 + \sum_{k=1}^n z (z-1)^{k-1} = 1 + \frac{z \left[ (z-1)^n - 1 \right]}{z-2}

for z > 2; for z=2, the lattice reduces to an infinite line and S_n = 2n + 1.^[14] This organization reflects the exponential growth characteristic of the Bethe lattice, where the size of generation n scales as (z-1)^{n-1} for large n, establishing a branching factor of z-1.^[14] In the infinite limit, the hierarchical structure ensures that the proportion of vertices in any finite set of generations approaches zero, though the lattice appears uniform when rerooted at any vertex due to its regularity.^[6]

Graph-theoretic aspects

Relation to Cayley trees and graphs

The Cayley tree is a finite rooted tree structure that approximates the Bethe lattice. In a Cayley tree with branching factor q \geq 1 (corresponding to coordination number z = q + 1) and n generations (or shells), the root vertex has degree q, while all non-leaf vertices in subsequent shells have degree q + 1, resulting in a total of N = 1 + \frac{q (q^n - 1)}{q - 1} vertices for q > 1. This finite graph features free boundary conditions at the outermost shell, making it a useful computational model in statistical mechanics where boundary effects are present but can be controlled.^[3] The Bethe lattice emerges as the thermodynamic limit of the Cayley tree when the number of generations n \to \infty, effectively removing boundary influences and yielding an infinite, regular tree where every vertex has exactly degree q + 1. In this limit, properties deep in the interior of the Cayley tree, far from the boundaries, match those of the Bethe lattice, allowing exact solutions in models like the Ising model by neglecting surface effects. Unlike the rooted Cayley tree, the Bethe lattice is unrooted, with all vertices equivalent, ensuring translational invariance across the structure.^[15] More broadly, the Bethe lattice relates to Cayley graphs, which are graphs constructed from a group and its generating set, connecting each group element to products with generators. Specifically, the Bethe lattice of even degree $2k is isomorphic to the undirected Cayley graph of the free group on k generators with respect to the free generating set, characterized by its acyclicity and regularity. However, general Cayley graphs may contain cycles if the underlying group has relations among generators, distinguishing them from the tree-like Bethe lattice, which remains cycle-free by construction.^[16] In physics literature, the terms "Bethe lattice" and "infinite Cayley tree" are sometimes used interchangeably, reflecting historical overlaps in applications to lattice models, though mathematicians emphasize the unrooted, boundary-free nature of the Bethe lattice to avoid confusion with finite approximations.^[3]

Random walks and return probabilities

The simple symmetric random walk on the Bethe lattice with coordination number q+1 proceeds by moving from any vertex to each of its q+1 neighbors with equal probability \frac{1}{q+1}.^[17] Due to the acyclic tree structure, the walk can only return to the origin after an even number of steps, so P_{2m+1}(0) = 0 for all m \geq 0, where P_n(0) denotes the probability of being at the origin after n steps.^[17] The return probabilities P_{2m}(0) satisfy the nonlinear recursion relation

P_{2m}(0) = \frac{q}{q+1} \sum_{k=0}^{m-1} P_{2k}(0) P_{2(m-1-k)}(0)

for m \geq 1, with initial condition P_0(0) = 1.^[17] This recursion arises from the tree geometry: after the first step to one of the q+1 neighbors (probability \frac{1}{q+1}), the walk effectively performs two independent excursions in subtrees before returning, but the factor \frac{q}{q+1} accounts for the biased exploration in the remaining q branches excluding the reverse direction. The generating function approach provides an exact closed-form solution. Let z = q+1 be the coordination number and \tilde{P}_{0,0}(\lambda) = \sum_{n=0}^\infty P_n(0) \lambda^n. Then,

\tilde{P}_{0,0}(\lambda) = \frac{2(z-1)/z}{(z-2)/z + \sqrt{1 - 4\lambda^2 (z-1)/z^2}},

valid for |\lambda| < 1/[z](/page/Z).^[17] The coefficients P_{2t}(0) can be extracted as

P_{2t}(0) = \frac{z-1}{z} \left( \frac{\sqrt{z-1}}{z} \right)^{2t} \frac{\Gamma(2t+1)}{\Gamma(t+1) \Gamma(t+2)} \, {}_2F_1\left( t + \frac{1}{2}, 1; t+2; \frac{4(z-1)}{z^2} \right),

where {}_2F_1 is the Gauss hypergeometric function.^[17] This involves binomial-like terms through the hypergeometric expansion, reflecting the combinatorial branching structure of paths returning to the origin.^[17] For large t, the asymptotic behavior is

P_{2t}(0) \sim \frac{2^{3/2} z (z-1)}{\sqrt{\pi} (z-2)^2} \, t^{-3/2} \exp\left( -t \ln \frac{z^2}{\sqrt{z-1}} \right).

^[17] The exponential decay confirms transience: for q \geq 2 (i.e., z \geq 3), the total sum \sum_{m=0}^\infty P_{2m}(0) = \tilde{P}_{0,0}(1) = \frac{z-1}{z-2} < \infty, so the walk visits the origin only finitely many times almost surely.^[17] This contrasts with recurrent simple random walks on one- and two-dimensional Euclidean lattices, where the sum diverges. The probability of ever returning to the origin is \frac{1}{z-1} < 1.^[17]

Number of closed walks

In graph theory, a closed walk of length n on the Bethe lattice is a sequence of n edges starting and ending at the same vertex, allowing revisits to vertices and edges. Since the Bethe lattice is an infinite, acyclic regular tree of degree z = q + 1 (with branching factor q \geq 1), all closed walks involve backtracking along paths. The number c_n of such closed walks starting and ending at a fixed vertex v equals the (v,v)-entry of the nth power of the adjacency matrix A, denoted (A^n)_{v,v}. Due to the vertex-transitivity of the Bethe lattice, this value is independent of the choice of v. For odd n, c_n = 0 because the graph is bipartite.^[18] For finite regular graphs, the total number of closed walks of length n is \operatorname{Tr}(A^n), and the average per vertex is \operatorname{Tr}(A^n)/|V|. In the infinite Bethe lattice, spectral theory provides the natural framework: c_n is the nth moment of the spectral measure \mu of the adjacency operator A with respect to the Dirac delta at v, given by

c_n = \int_{-2\sqrt{q}}^{2\sqrt{q}} \lambda^n \, d\mu(\lambda).

The support of \mu is the interval [-2\sqrt{q}, 2\sqrt{q}], and \mu has the absolutely continuous Kesten-McKay distribution with density

\rho(\lambda) = \frac{q+1}{2\pi} \frac{\sqrt{4q - \lambda^2}}{(q+1)^2 - \lambda^2}, \quad |\lambda| < 2\sqrt{q}.

Thus,

c_n = \int_{-2\sqrt{q}}^{2\sqrt{q}} \lambda^n \rho(\lambda) \, d\lambda.

This distribution, originally derived by Kesten for symmetric random walks on groups including trees and extended by McKay to random regular graphs, characterizes the limiting spectral behavior of the Bethe lattice.^[18] Due to the self-similar tree structure, c_n also satisfies an exact recurrence relation derived from branching. Consider walks starting at v: the first step goes to one of the q+1 neighbors u, and the remaining n-1 steps must return from u to v. Let u_m denote the number of walks of length m from a neighbor u back to v. Then c_n = (q+1) u_{n-1}. For u_m, the first step from u is either directly back to v (if m=1) or to one of the q subtree roots w (excluding the direction to v), followed by a closed walk of length m-2 at w in its subtree (which is isomorphic to the original Bethe lattice), and then back to u. This yields the coupled recurrence u_m = \delta_{m,1} + q \, c_{m-2} for m \geq 2, with u_0 = 0, leading to the closed form c_n = (q+1) u_{n-1} solvable iteratively. Generating functions for these quantities satisfy algebraic equations reflecting the branching, enabling explicit computation.^[19] Asymptotically, for large even n = 2k, the growth of c_n is dominated by the largest "eigenvalue" at the spectral edge $2\sqrt{q}, yielding c_n \sim \kappa (2\sqrt{q})^n n^{-3/2} for some constant \kappa > 0 depending on q. This exponential growth rate $2\sqrt{q} equals the spectral radius of A, highlighting the expansive nature of walks on the tree before backtracking closes them.^[18]

Geometric and algebraic interpretations

Hyperbolic geometry

The Bethe lattice with coordination number z \geq 2 admits an isometric embedding into the hyperbolic plane \mathbb{H}^2 equipped with constant Gaussian curvature -1, where the graph's edges are realized as geodesic segments of fixed length satisfying the hyperbolic metric.^[20] This embedding represents the Bethe lattice as the limiting case of a regular tiling denoted \{ \infty, z \} in hyperbolic geometry, in which vertices correspond to points in \mathbb{H}^2 and edges to geodesics, preserving distances and the tree-like structure without cycles.^[20] The conformal Poincaré disk model is commonly used for this realization, mapping the lattice inside a unit disk with the hyperbolic metric ds^2 = 4 \frac{dr^2 + r^2 d\theta^2}{(1 - r^2)^2}.^[21] In this embedding, the graph distance n between vertices corresponds to the hyperbolic geodesic distance d, which grows linearly with n for adjacent vertices but accounts for the branching structure at larger scales. Specifically, the distance from a central vertex to those in the nth generation aligns with radial coordinates in \mathbb{H}^2, where successive generations occupy annuli whose widths reflect the constant edge length.^[19] The geodesic distance between nearest neighbors remains uniform throughout the lattice, enabling a direct mapping of the discrete metric to the continuous hyperbolic one.^[20] The exponential volume growth of the Bethe lattice—where the number of vertices up to graph distance n scales as (z-1)^n for large n—precisely mirrors the area growth in \mathbb{H}^2, which expands as e^r with radius r proportional to n. This equivalence arises from the negative curvature, ensuring that the embedded lattice's site density follows the hyperbolic volume element dV \propto \sinh r \, dr \, d\theta.^[19] Such growth distinguishes the Bethe lattice from Euclidean lattices, highlighting its suitability for modeling systems with unbounded expansion. The embedding extends to a compactification in the Poincaré disk, where the lattice fills the interior, and its infinite branches approach the boundary circle at infinity, known as the ideal boundary or conformal boundary of \mathbb{H}^2. Each infinite ray in the tree converges to a distinct ideal point on this boundary, parameterizing the space of ends of the lattice.^[21] This structure underscores the Bethe lattice's role as an approximation to finite hyperbolic tilings or as a discrete model for subgroups of isometries in the hyperbolic plane, facilitating studies of geometric limits and fractal-like boundaries.^[20]

Lattices in Lie groups

For even coordination number z = 2n, the Bethe lattice is the undirected Cayley graph of the free group F_n with respect to its standard symmetric generating set consisting of the n generators and their inverses, providing a discrete model for the action of free subgroups within semisimple Lie groups of rank one, such as \mathrm{SL}(2, \mathbb{R}). Schottky groups, which are freely generated discrete subgroups of \mathrm{PSL}(2, \mathbb{R}), exemplify this connection: their Cayley graphs are precisely the z-regular Bethe lattices with z = 2g for genus g, capturing the tree-like geometry of their free actions without fixed points. These groups serve as building blocks for more general lattices, as finite-index subgroups of Fuchsian lattices in \mathrm{SL}(2, \mathbb{R}) often decompose into free products involving such Schottky components, reflecting the underlying hyperbolic structure.^[22] In the non-Archimedean setting, the Bethe lattice emerges directly as the Bruhat-Tits building associated to \mathrm{SL}(2, K), where K is a local field such as \mathbb{Q}_p; this building is a z-regular tree with z = p+1, with vertices corresponding to homothety classes of \mathcal{O}_K-lattices in K^2 and edges linking contained lattices differing by the uniformizer. The group \mathrm{SL}(2, K) acts transitively on the edges of this tree, and discrete subgroups, including lattices like \mathrm{SL}(2, \mathcal{O}_K), produce quotients that are finite graphs for uniform (cocompact) lattices or finite cores with attached infinite trees at cusps for non-uniform ones. This construction extends to higher-rank analogs via products of such trees, and to real groups like \mathrm{SO}(n,1), where discrete subgroups yield tree-like fundamental domains in the boundary at infinity of hyperbolic space \mathbb{H}^n, analogous to the p-adic case. For instance, with coordination number z=3 (corresponding to p=2), the Bruhat-Tits tree for \mathrm{SL}(2, \mathbb{Q}_2) is the 3-regular Bethe lattice, on which arithmetic lattices act with finite-volume quotients.^[23] Arithmetic lattices in these groups, such as those arising from quaternion algebras over number fields, exhibit properties like the absence of the congruence subgroup property, leading to dense orbits on the tree and non-arithmetic examples. Thin discrete subgroups—those with infinite-index normal subgroups—generate particularly tree-like fundamental domains, consisting of a compact core with infinite regular trees emanating from boundary points, mirroring the Bethe lattice's infinite branching. In higher dimensions, lattices in groups like \mathrm{SO}(n,1) for n \geq 2 produce similar structures, where the fundamental domain unfolds into a tree via the action on horospheres. In representation theory, the Bethe lattice's symmetry under group actions enables the construction of Hecke operators as adjacency matrices on the tree, facilitating the study of spherical representations of p-adic groups like \mathrm{SL}(2, \mathbb{Q}_p). Automorphic forms on these trees are defined via \Gamma-invariant harmonic cocycles or improper Eisenstein series, providing analogs to classical modular forms and encoding arithmetic data through the tree's quotient; for example, p-adic automorphic forms arise from quaternion orders acting on the Bruhat-Tits tree, with fundamental domains computable for Shimura curves. This framework links to broader automorphic representation theory, where tree actions yield explicit models for unitary representations and trace formulas.^[24]^[25]^[26]

Applications in statistical mechanics

Ising model solutions

The ferromagnetic Ising model on the Bethe lattice is a paradigmatic example of an exactly solvable system in statistical mechanics, owing to the acyclic tree structure that permits a recursive computation of the partition function. The model consists of Ising spins \sigma_i = \pm 1 at each vertex of the Bethe lattice with coordination number q \geq 2, governed by the Hamiltonian

H = -J \sum_{\langle i,j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i,

where J > 0 denotes the ferromagnetic nearest-neighbor coupling, h is the uniform external magnetic field, \langle i,j \rangle runs over edges, and the sums are over all vertices and edges, respectively.^[2] The exact solution employs a recursion over finite-depth subtrees rooted at a vertex, leveraging the self-similar structure of the lattice. For a subtree of depth n, define the partial partition functions g_n^\sigma as the sum over configurations of the subtree with the root spin fixed to \sigma = \pm 1, in the presence of the external field h. These satisfy the recursion relations

g_n^+ = e^{\beta h} \left( e^{\beta J} g_{n-1}^+ + e^{-\beta J} g_{n-1}^- \right)^{q-1},

g_n^- = e^{-\beta h} \left( e^{-\beta J} g_{n-1}^+ + e^{\beta J} g_{n-1}^- \right)^{q-1},

with initial conditions g_0^\pm = 1. Equivalently, the recursion can be formulated in terms of an effective field via the cavity method, where the message or bias \lambda_n = \tanh(\beta h_n) from a subtree of depth n obeys a nonlinear map of the form \lambda_n = \tanh\left(\beta J + \mathrm{atanh}\left( \tanh(\beta J) \lambda_{n-1} \right)\right) for the contribution from a single subbranch, iterated over the q-1 branches at each level to yield the full effective field at the root. In the thermodynamic limit n \to \infty, the fixed point of this recursion determines the local magnetization m = \tanh(\beta h_\infty).^[27]^[2] In zero external field (h = 0), the recursion simplifies, and the ratio x_n = g_n^- / g_n^+ follows

x_n = \left[ \frac{e^{-\beta J} + e^{\beta J} x_{n-1}}{e^{\beta J} + e^{-\beta J} x_{n-1}} \right]^{q-1},

with x_0 = 1. The infinite-depth limit x_\infty solves the fixed-point equation x = \left[ \frac{e^{-\beta J} + e^{\beta J} x}{e^{\beta J} + e^{-\beta J} x} \right]^{q-1}, yielding the spontaneous magnetization m = (1 - x_\infty)/(1 + x_\infty). The partition function for a finite Cayley tree of depth n (approximating the lattice) is Z = (g_n^+)^q + (g_n^-)^q, constructed from a central vertex connected to q identical branches. In the infinite-volume limit, the free energy per site is

-\beta f = \frac{1}{q} \log \left[ (g_\infty^+)^q + (g_\infty^-)^q \right] + \left(1 - \frac{1}{q}\right) \log (g_\infty^+ + g_\infty^-),

where g_\infty^\pm are the fixed-point values normalized such that g_\infty^- = 1 and g_\infty^+ = 1/x_\infty; this captures the bulk thermodynamics by balancing the central-site and branching contributions.^[27]^[2] The model exhibits a second-order phase transition from a paramagnetic phase (m = 0) at high temperatures to a ferromagnetic phase (m > 0) at low temperatures, with the critical inverse temperature \beta_c determined by the instability of the trivial fixed point x = 1, giving \tanh(\beta_c J) = 1/(q-1) or k_B T_c / J = 1 / \artanh[1/(q-1)]. For example, with q=3, T_c \approx 1.82 J / k_B; as q \to \infty, T_c \to q J / k_B, recovering the mean-field limit. The phase diagram in the (\beta J, h) plane features a line of second-order transitions ending at a critical point for h \neq 0, but the zero-field case highlights the pure ferromagnetic transition enabled by the loop-free geometry, which avoids the approximations needed for periodic lattices.^[2]^[27]

Anderson model and localization

The Anderson model on the Bethe lattice describes non-interacting electrons in a disordered tight-binding system, where the lattice's tree-like structure without loops allows for exact treatments of quantum hopping and localization phenomena. The Hamiltonian is given by

H = \sum_i \epsilon_i |i\rangle\langle i| + t \sum_{\langle i j \rangle} \left( |i\rangle\langle j| + |j\rangle\langle i| \right),

with on-site random energies \epsilon_i drawn from a uniform distribution over [-W/2, W/2] and hopping amplitude t (often set to 1) between nearest neighbors on the Bethe lattice of coordination number q \geq 3.^[28] Exact solutions for the model's spectral properties rely on self-consistent equations derived from the recursive structure of the Bethe lattice, which map the problem to a distribution of cavity Green's functions. The local Green's function G(z) at complex energy z satisfies the distributional equation

G(z) \stackrel{d}{=} \frac{1}{z - \epsilon - t^2 \sum_{m=1}^q \hat{G}_m(z)},

where \epsilon is random, and the \hat{G}_m(z) are i.i.d. copies of a related cavity Green's function \hat{G}(z) \stackrel{d}{=} 1/(z - \epsilon - t^2 \sum_{m=1}^{q-1} \hat{G}_m(z)); these equations, first formulated in a self-consistent theory, enable numerical solution for the averaged Green's function and thus the density of states \rho(E) = -\frac{1}{\pi} \operatorname{Im} \langle G(E + i0^+) \rangle.^[29]^[28] The density of states exhibits a band structure influenced by disorder, with edges roughly at \pm (2t\sqrt{q-1} + W/2), and for weak disorder, it approximates the Kesten-McKay semicircle law of the clean model but develops tails and gaps as W increases. In the infinite-coordination limit (q \to \infty), rescaling t^2 q \to 1 yields a metallic phase with no localization, analogous to infinite-dimensional dynamics where disorder does not localize states. For finite q, an Anderson localization transition occurs at a critical disorder strength W_c, beyond which all states localize; numerical solutions give W_c \approx 17.4 t for q=3, increasing asymptotically as W_c \approx 4.7 q \ln q \, t for large q, with the precise value tied to the stability of the Kesten distribution for the continued-fraction representation of Green's functions.^[29]^[28] A key feature is the mobility edge, an energy E_c(W) separating extended (ergodic) states near the band center from localized states near the edges, absent in low-dimensional lattices but present here due to the lattice's hyperbolic geometry; for example, at W = 12 t and q=3, E_c \approx 4 t. Near the band edges, Lifshitz tails emerge in \rho(E), arising from rare regions of atypically low disorder that support bound states, with exponential decay \rho(E) \sim \exp\left[ -c q^{1/4} (E - E_{\min})^{-1/2} \ln(W / (E - E_{\min})) \right] for E \to E_{\min}^+, exactly computable via large-deviation analysis on the tree.^[28]^[12]

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Exact clesed form of the return probability on the Bethe lattice - arXiv
Nov 28, 1994 · An exact closed form solution for the return probability of a random walk on the Bethe lattice is given. The long-time asymptotic form confirms ...Missing: formula | Show results with:formula
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[PDF] arXiv:2009.11950v2 [math.PR] 7 Jun 2021
Jun 7, 2021 · This paper connects closed walks on a regular tree, the Kesten-McKay law, and standard Young tableaux, showing the moments of the law are given ...Missing: Bethe | Show results with:Bethe
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[PDF] random walk on the bethe lattice and hyperbolic brownian motion
Abstract. We give the exact solution to the problem of a random walk on the Bethe lattice through a mapping on an asymmetric random walk on the half-line.Missing: \frac
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[PDF] LE JOURNAL DE PHYSIQUE - LETTRES - HAL
The Bethe lattice is usually defined by the fact that it does not include any closed ... - We have shown in this paper that every Bethe lattice of coordination ...
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[PDF] Metric properties of the Bethe lattice and the Husimi cactus - HAL
Feb 4, 2008 · distances can thus be measured with the help of non-Euclidean hyperbolic geometry. We use barycentric coordinates. [13] on the projective ...
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[PDF] Schottky Groups and Limit Sets Introduction to Fractal Geometry and ...
Fuchsian Schottky group: Schottky group in SL2(R). (preserves upper and ... are Fuchsian groups, Gi ⊂ PSL(2,R) group ˜Γ ⊂ Γ is itself discrete purely ...
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[PDF] Essays in harmonic analysis on p-adic SL(2) Geometry of the tree
Apr 10, 2019 · The Bruhat-Tits tree of G = SL2(k) is a graph X on which the group PGL2(k) acts. The geometry of this graph encodes much of the group structure.Missing: Bethe | Show results with:Bethe
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https://arxiv.org/abs/1201.0356
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Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp ...
Jan 1, 2012 · Abstract:We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp).
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Improper Eisenstein Series on Bruhat-Tits Trees. - EuDML
... automorphic forms. 11F12: Automorphic forms, one variable. Arithmetic algebraic geometry (Diophantine geometry). 11G09: Drinfeld modules; higher-dimensional ...
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### Summary of Ising Model on Bethe Lattice (Ferromagnetic Case, Zero Field)
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[1005.0342] Anderson model on Bethe lattices: density of states ...
May 3, 2010 · We revisit the Anderson localization problem on Bethe lattices, putting in contact various aspects which have been previously only discussed separately.
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A selfconsistent theory of localization - IOPscience
The equations used are exact for a Bethe lattice. The selfconsistency condition gives a nonlinear integral equation in two variables for the probability ...